Generalization of an inequality of Alzer for negative powers

Authors

  • Chao-Ping Chen
  • Feng Qi

DOI:

https://doi.org/10.5556/j.tkjm.36.2005.113

Abstract

Let $ \{a_n \}_{n=1}^{\infty} $ be a positive, strictly increasing, and logarithmically concave sequence satisfying $ (a_{n+1}/a_{n})^{n}<(a_{n+2}/a_{n+1})^{n+1} $. Then we have

$$\frac {a_n}{a_{n+m}}<\left(\frac {1}{n} \sum_{i=1}^{n}a_{i}^r\bigg/ \frac {1}{n+m} \sum_{i=1}^{n+m}a_{i}^r \right)^{1/r}, $$

where $ n $, $ m $ are natural numbers and $ r $ is a positive real number. The lower bound is the best possible. This generalizes an inequality of Alzer for negative powers.

Author Biographies

Chao-Ping Chen

Department of Applied Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan 454010, CHINA.

Feng Qi

Department of Applied Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan 454010, CHINA.

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Published

2005-09-30

How to Cite

Chen, C.-P., & Qi, F. (2005). Generalization of an inequality of Alzer for negative powers. Tamkang Journal of Mathematics, 36(3), 219-222. https://doi.org/10.5556/j.tkjm.36.2005.113

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Papers

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